2 weeks ago, Samir Siksek https://arxiv.org/abs/1505.00647 proved the more than 150-years-old conjecture that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, 454 is a sum of at most seven positive cubes. It was known long ago that every sufficiently large integer is a sum of at most seven positive cubes, but the explicit list of exceptions was not known.
My question is about 5th powers.
a) For which k do we know the full list of exceptions not representable as a sum of k 5th powers?
b) For which k do we at least have effective bound $N_k$, such that every $n \geq N_k$ is representable? (Here, the interesting values are $17\leq k\leq 36$, because the list of exceptions is known to be empty for $k=37$, and not even known to be finite for $k\leq 16$).
c) Do we know an efficient (possibly randomized or based on unproved conjectures) algorithm to check whether a fixed large number n is representable as sum of $k$ 5th powers (for fixed $2\leq k\leq 36$)?
